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| Mirrors > Home > MPE Home > Th. List > Mathboxes > f1omo | Structured version Visualization version GIF version | ||
| Description: There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 48797 assuming ax-un 7756 (see f1omoALT 48799). (Contributed by Zhi Wang, 19-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| f1omo.1 | ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) | 
| Ref | Expression | 
|---|---|
| f1omo | ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 1oex 8517 | . . . 4 ⊢ 1o ∈ V | |
| 2 | eqid 2736 | . . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋) | |
| 3 | 1, 2 | fvconst0ci 48796 | . . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) | 
| 4 | mo0 48738 | . . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) | |
| 5 | el1o 8534 | . . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅) | |
| 6 | el1o 8534 | . . . . . . . 8 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅) | |
| 7 | eqtr3 2762 | . . . . . . . 8 ⊢ ((𝑦 = ∅ ∧ 𝑥 = ∅) → 𝑦 = 𝑥) | |
| 8 | 5, 6, 7 | syl2anb 598 | . . . . . . 7 ⊢ ((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥) | 
| 9 | 8 | gen2 1795 | . . . . . 6 ⊢ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥) | 
| 10 | eleq1w 2823 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 1o ↔ 𝑥 ∈ 1o)) | |
| 11 | 10 | mo4 2565 | . . . . . 6 ⊢ (∃*𝑦 𝑦 ∈ 1o ↔ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)) | 
| 12 | 9, 11 | mpbir 231 | . . . . 5 ⊢ ∃*𝑦 𝑦 ∈ 1o | 
| 13 | eleq2w2 2732 | . . . . . 6 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ 𝑦 ∈ 1o)) | |
| 14 | 13 | mobidv 2548 | . . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ ∃*𝑦 𝑦 ∈ 1o)) | 
| 15 | 12, 14 | mpbiri 258 | . . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) | 
| 16 | 4, 15 | jaoi 857 | . . 3 ⊢ ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)) | 
| 17 | 3, 16 | ax-mp 5 | . 2 ⊢ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) | 
| 18 | f1omo.1 | . . . . 5 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o})) | |
| 19 | 18 | fveq1d 6907 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋)) | 
| 20 | 19 | eleq2d 2826 | . . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋))) | 
| 21 | 20 | mobidv 2548 | . 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))) | 
| 22 | 17, 21 | mpbiri 258 | 1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∀wal 1537 = wceq 1539 ∈ wcel 2107 ∃*wmo 2537 ∅c0 4332 {csn 4625 × cxp 5682 ‘cfv 6560 1oc1o 8500 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-1o 8507 | 
| This theorem is referenced by: indthinc 49134 prsthinc 49136 | 
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